Fermat’s and energy variation principles in field theory

In general relativity the light is assumed to propagate in the vacuum along null geodesic in a pseudo-Riemannian manifold. Besides the geodesics principle in a classical field theory there exists the Fermat's principle for stationary gravity fields.^[1]

Fermat's principle

In more general case for conformally stationary spacetime ^[2] with coordinates ${\displaystyle (t,x^{1},x^{2},x^{3})}$ a Fermat metric takes form

${\displaystyle g=e^{2f(t,x)}[(dt+\phi _{\alpha }(x)dx^{\alpha })^{2}-{\hat {g}}_{\alpha \beta }dx^{\alpha }dx^{\beta }]}$,

where conformal factor ${\displaystyle f(t,x)}$ depending on time $t$ and space coordinates ${\displaystyle x^{\alpha }}$ does not affect the lightlike geodesics apart from their parametrization.

Fermat's principle for a pseudo-Riemannian manifold states that the light ray path between points ${\displaystyle x_{a}=(x_{a}^{1},x_{a}^{2},x_{a}^{3})}$ and ${\displaystyle x_{b}=(x_{b}^{1},x_{b}^{2},x_{b}^{3})}$ corresponds to zero variation of action

${\displaystyle S=\int _{\mu _{b}}^{\mu _{a}}\left({\sqrt {{\hat {g}}_{\alpha \beta }{\frac {dx^{\alpha }}{d\mu }}{\frac {dx^{\beta }}{d\mu }}}}+\phi _{\alpha }(x){\frac {dx^{\alpha }}{d\mu }}\right)d\mu }$,

where $\mu$ is any parameter ranging over an interval ${\displaystyle [\mu _{a},\mu _{b}]}$ and varying along curve with fixed endpoints ${\displaystyle x_{a}=x(\mu _{a})}$ and ${\displaystyle x_{b}=x(\mu _{b})}$.

Principle of stationary integral of energy

In principle of stationary integral of energy for a light-like particle's motion, the pseudo-Riemannian metric with coefficients ${\displaystyle {\tilde {g}}_{ij}}$ is defined by a transformation

${\displaystyle {\tilde {g}}_{00}=\rho ^{2}{g}_{00},\,\,\,\,{\tilde {g}}_{0k}=\rho {g}_{0k},\,\,\,\,{\tilde {g}}_{kq}={g}_{kq}.}$

With time coordinate $x^{0}$ and space coordinates with indexes k,q=1,2,3 the line element is written in form

${\displaystyle ds^{2}=\rho ^{2}g_{00}(dx^{0})^{2}+2\rho g_{0k}dx^{0}dx^{k}+g_{kq}dx^{k}dx^{q},}$

where $\rho$ is some quantity, which is assumed equal 1 and regarded as the energy of the light-like particle with ${\displaystyle ds=0}$. Solving this equation for $\rho$ under condition ${\displaystyle g_{00}\neq 0}$ gives two solutions

${\displaystyle \rho ={\frac {-g_{0k}v^{k}\pm {\sqrt {(g_{0k}g_{0q}-g_{00}g_{kq})v^{k}v^{q}}}}{g_{00}v^{0}}},}$

where ${\displaystyle v^{i}=dx^{i}/d\mu }$ are elements of the four-velocity. Even if one solution, in accordance with making definitions, is ${\displaystyle \rho =1}$.

With ${\displaystyle g_{00}=0}$ and ${\displaystyle g_{0k}\neq 0}$ even if for one k the energy takes form

${\displaystyle \rho =-{\frac {g_{kq}v^{k}v^{q}}{2v_{0}v^{0}}}.}$

In both cases for the free moving particle the lagrangian is

${\displaystyle L=-\rho .}$

Its partial derivatives give the canonical momenta

${\displaystyle p_{\lambda }={\frac {\partial L}{\partial v^{\lambda }}}={\frac {v_{\lambda }}{v^{0}v_{0}}}}$

and the forces

${\displaystyle F_{\lambda }={\frac {\partial L}{\partial x^{\lambda }}}={\frac {1}{2v^{0}v_{0}}}{\frac {\partial g_{ij}}{\partial x^{\lambda }}}v^{i}v^{j}.}$

Momenta satisfy energy condition ^[3] for closed system

${\displaystyle \rho =v^{\lambda }p_{\lambda }-L.}$

Standard variational procedure is applied to action

${\displaystyle S=\int _{\mu _{b}}^{\mu _{a}}Ld\mu =-\int _{\mu _{b}}^{\mu _{a}}\rho d\mu ,}$

which is integral of energy. Stationary action is conditional upon zero variational derivatives δS/δx^λ and leads to Euler–Lagrange equations

${\displaystyle {\frac {d}{d\mu }}{\frac {\partial \rho }{\partial v^{\lambda }}}-{\frac {\partial \rho }{\partial x^{\lambda }}}=0,}$

which is rewritten in form

${\displaystyle {\frac {d}{d\mu }}p_{\lambda }-F_{\lambda }=0.}$

After substitution of canonical momentum and forces they give motion equations of lightlike particle in a free space

${\displaystyle {\frac {dv^{0}}{d\mu }}+{\frac {v^{0}}{2v_{0}}}{\frac {\partial g_{ij}}{\partial x^{0}}}v^{i}v^{j}=0}$

and

${\displaystyle (g_{k\lambda }v_{0}-g_{0k}v_{\lambda }){\frac {dv^{k}}{d\mu }}+\left[{\frac {1}{2v_{0}}}{\frac {\partial g_{ij}}{\partial x^{0}}}(g_{00}v^{0}v_{\lambda }+g_{k\lambda }v^{k}v_{0})-{\frac {1}{2}}{\frac {\partial g_{ij}}{\partial x^{\lambda }}}v_{0}+{\frac {\partial g_{i\lambda }}{\partial x^{j}}}v_{0}-{\frac {\partial g_{0i}}{\partial x^{j}}}v_{\lambda }\right]v^{i}v^{j}=0.}$

Static spacetime

For the static spacetime the first equation of motion with appropriate parameter $\mu$ gives ${\displaystyle v^{0}=1}$ . Canonical momentum and forces will be

${\displaystyle p_{\lambda }={\frac {v_{\lambda }}{g_{00}}};\qquad F_{\lambda }={\frac {1}{2g_{00}}}{\frac {\partial g_{ij}}{\partial x^{\lambda }}}v^{i}v^{j}}$.

For the isotropic paths a transformation to metric ${\displaystyle {\overline {g}}_{ij}={\frac {g_{ij}}{g_{00}}}}$ is equivalent to replacement of parameter $\mu$ on ${\displaystyle d{\overline {\mu }}={\frac {d\mu }{\sqrt {g_{00}}}}}$. The curve of motion of lightlike particle in four-dimensional spacetime and value of energy $\rho$ are invariant under this reparametrization. Canonical momentum and forces take form

${\displaystyle {\overline {p}}_{\lambda }={\frac {{\overline {v}}_{\lambda }}{{\overline {g}}_{00}}};\qquad {\overline {F}}_{\lambda }={\frac {1}{2}}{\frac {\partial {\overline {g}}_{ij}}{\partial x^{\lambda }}}{\overline {v}}^{i}{\overline {v}}^{j}.}$

Substitution of them in Euler–Lagrange equations gives

${\displaystyle {\frac {d}{d\mu }}\left({\overline {g}}_{\lambda k}{\overline {v}}^{k}\right)={\frac {1}{2}}{\frac {\partial {\overline {g}}_{ij}}{\partial x^{\lambda }}}{\overline {v}}^{i}{\overline {v}}^{j}}$.

This expression after some calculation becomes null geodesic equations

${\displaystyle {\frac {d^{2}x^{\lambda }}{d\mu ^{2}}}+\Gamma _{ij}^{\lambda }{\frac {dx^{i}}{d\mu }}{\frac {dx^{j}}{d\mu }}=0,}$

where ${\displaystyle \Gamma _{ij}^{\lambda }}$ are the second kind Christoffel symbols with respect to the given metric tensor.

So in case of the static spacetime the geodesic principle and the energy variational method as well as Fermat's principle give the same solution for the light propagation.

See also

• Fermat's principle

References

1. Landau, Lev D.; Lifshitz, Evgeny F. (1980), The Classical Theory of Fields (4th ed.), London: Butterworth-Heinemann, p. 273, ISBN 9780750627689
2. Perlik, Volker (2004), "Gravitational Lensing from a Spacetime Perspective", Living Rev. Relativity, 7 (9), Chapter 4.2
3. Landau, Lev D.; Lifshitz, Evgeny F. (1976), Mechanics Vol. 1 (3rd ed.), London: Butterworth-Heinemann, p. 14, ISBN 9780750628969